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Sum of inverse squares of natural numbers
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed
Basel_problem
City in Switzerland
the so-called "Basel Problem" was solved in the city, which is regarded as one of the most important achievements in mathematics. Basel was ranked the
Basel
Number, approximately 3.14
simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it
Pi
Negative integer two units from the origin in mathematics
positive; the inverse-square law; grid turbulence decay; and the Basel problem. The Basel problem states that the sum of the square reciprocals of natural numbers
−2
Summation formula
}}+\int _{m}^{n}f^{(7)}(x){\frac {P_{7}(x)}{7!}}\,dx.\end{aligned}}} The Basel problem is to determine the sum 1 + 1 4 + 1 9 + 1 16 + 1 25 + ⋯ = ∑ n = 1 ∞
Euler–Maclaurin_formula
Swiss mathematician (1707–1783)
accomplishments, solving several unsolved problems in number theory and analysis, including the famous Basel problem. Euler has also been credited for discovering
Leonhard_Euler
Analytic function in mathematics
ζ ( 2 ) {\displaystyle \zeta (2)} , provides a solution to the Basel problem. In 1979, Roger Apéry proved the irrationality of ζ(3), and as a consequence
Riemann_zeta_function
Problem in probability theory
{1}{1^{2}}}+{\frac {1}{2^{2}}}+\cdots +{\frac {1}{n^{2}}}+\cdots } (see Basel problem). Bound the desired probability using the Chebyshev inequality: P
Coupon_collector's_problem
"Euler's number"). The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number π2 /6, or ζ(2) where ζ is the Riemann
List_of_sums_of_reciprocals
Array of line segments normal to points of a square lattice
{x}{x+y}},{\frac {y}{x+y}},{\frac {1}{x+y}}\right).} The solution to the Basel problem can be used to show that the proportion of points in the n × n {\displaystyle
Euclid's_orchard
Number divisible only by 1 and itself
Leonhard Euler and his first major result, the solution to the Basel problem. The problem asked for the value of the infinite sum 1 + 1 4 + 1 9 + 1 16 +
Prime_number
Topics referred to by the same term
waste between nations Basel problem, a famous problem in number theory Basil, a plant and culinary herb Basil (name), of which Basel is a variant Basal (disambiguation)
Basel_(disambiguation)
Index of articles associated with the same name
the squares of the first three equals the square of the fourth. The Basel problem, solved by Euler in terms of π {\displaystyle \pi } , asked for an exact
Sum_of_squares
Banking regulation framework
Basel III is a framework published by the Basel Committee on Banking Supervision (BCBS) in response to the deficiencies in financial regulation revealed
Basel_III
Approximations of π Arithmetic–geometric mean Bailey–Borwein–Plouffe formula Basel problem Borwein's algorithm Buffon's needle Cadaeic Cadenza Chronology of computation
List_of_topics_related_to_π
1897 proposed law to define squaring the circle
is not considered a solution to the ancient problem of squaring the circle. This is because the problem is to construct the area using a compass and
Indiana_pi_bill
Infinite series with alternating signs
n = 0 respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as
1_−_2_+_3_−_4_+_⋯
.. 1/6 One sixth. Often appears in mathematical equations, such as in the sum of squares of the integers and in the solution to the Basel problem.
List_of_numbers
Perimeter of a circle or ellipse
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Circumference
Uses of the constant
{1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}} (see also Basel problem and Riemann zeta function) ζ ( 4 ) = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 +
List_of_formulae_involving_π
Indian mathematician-astronomer (476–550)
longer used as the function versin has become obsolete.[citation needed] A problem of great interest to Indian mathematicians since ancient times has been
Aryabhata
Greek mathematician and physicist (c. 287 – 212 BC)
During the Renaissance, the Editio princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin
Archimedes
Decomposition of periodic functions
method of solving the heat problem was made possible by Fourier's work. Another application is to solve the Basel problem by using Parseval's theorem
Fourier_series
π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6} is the solution to the Basel problem, the Airy zeta function may be exactly evaluated at s = 2: ζ A i ( 2
Airy_zeta_function
French mathematician (1916–1994)
{1}{27}}+{\frac {1}{64}}+\cdots \neq {\frac {p}{q}}} Apéry's constant Basel problem Apéry, François (1996). "Roger Apéry, 1916-1994: A Radical Mathematician"
Roger_Apéry
Constants of the mathematical zeta function
The computation of ζ ( 2 ) {\displaystyle \zeta (2)} is known as the Basel problem. The value of ζ ( 4 ) {\displaystyle \zeta (4)} is related to the Stefan–Boltzmann
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
{(-1)^{n}z^{2n+1}}{2n+1}}.} His use of power series enabled him to solve the famous Basel problem in 1735: lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle
Contributions of Leonhard Euler to mathematics
Contributions_of_Leonhard_Euler_to_mathematics
Concept in geometry
pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence
Area_of_a_circle
Welsh mathematician (1675–1749)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
William_Jones_(mathematician)
Signed odd unit fractions sum to π/4
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Leibniz_formula_for_π
1998 mathematics book by Aigner and Ziegler
\pi } (pi) is irrational. Chapter 9: Four proofs of the solution to Basel problem, namely that ∑ n ≥ 1 ∞ 1 n 2 = π 2 6 {\displaystyle \sum _{n\geq 1}^{\infty
Proofs_from_THE_BOOK
Two numbers without shared prime factors
example of an Euler product, and the evaluation of ζ(2) as π2/6 is the Basel problem, solved by Leonhard Euler in 1735. There is no way to choose a positive
Coprime_integers
Concept of complex analysis
}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}} which is a proof of the Basel problem. The same argument works for all f ( x ) = x − 2 n {\displaystyle f(x)=x^{-2n}}
Residue_theorem
Special mathematical function defined as sin(x)/x
this series to the expansion of the infinite product form to solve the Basel problem. The product of 1-D sinc functions readily provides a multivariate sinc
Sinc_function
Value approached by a mathematical object
Otherwise, the series is said to be divergent. A classic example is the Basel problem, where a n = 1 / n 2 {\displaystyle a_{n}=1/n^{2}} . Then ∑ n = 1 ∞
Limit_(mathematics)
Hobson, Ernest William (1913). 'Squaring the Circle': a History of the Problem (PDF). Cambridge University Press. p. 27. Yoshio, Mikami; Eugene Smith
Chronology of computation of pi
Chronology_of_computation_of_pi
Type of figurate number
polygonal number Polyhedral number Fermat polygonal number theorem See Basel problem. Tattersall, James J. (2005). Elementary Number Theory in Nine Chapters
Polygonal_number
Annual mathematical celebration on March 14
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Pi_Day
Problem of constructing equal-area shapes
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given
Squaring_the_circle
Difference between logarithm and harmonic series
series of inverse triangular numbers also features in the study of the Basel problem posed by Pietro Mengoli. Mengoli proved that ∑ k = 1 ∞ 1 2 T k = 1 {\displaystyle
Euler's_constant
Book by Hiroshi Yūki
independently come up with an idea that leads to an elegant solution to the Basel problem, they become close friends. She has a deep knowledge of mathematics
Math_Girls
Swiss 14th-century pogrom
The Basel Massacre was an anti-Semitic massacre in Basel, which occurred in 1349 in connection with alleged well poisoning as part of the Black Death persecutions
Basel_Massacre
Fixed number that has received a name
and elementary functions originated when Euler famously solved the Basel problem by giving ζ ( 2 ) = 1 6 π 2 {\displaystyle \zeta (2)={\frac {1}{6}}\pi
Mathematical_constant
Divergent sum of positive unit fractions
π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6} , the solution to the Basel problem, Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} , proved by Roger
Harmonic_series_(mathematics)
Set of integers whose sum of reciprocals diverges
2, 3, ...) is a large set. The set of square numbers is small (see Basel problem § The Riemann zeta function). So is the set of cube numbers, the set
Large_set_(combinatorics)
Infinite products of functions indexed by primes
\zeta (2)={\frac {\pi ^{2}}{6}}} by the previous product, known as Basel problem, one finds that π 2 = ( ∏ p ≡ 1 ( mod 4 ) p p + 1 ) ( ∏ p ≡ 3 ( mod
Euler_product
Conditions for switching order of integration in calculus
g(x)\,g(xy)\,\mathrm {d} x\,\mathrm {d} y} This can be applied to the Basel problem: π 2 4 = ( arcsin ( 1 ) ) 2 = [ ∫ 0 1 1 1 − x 2 d x ] 2 = ∫ 0 1 ∫
Fubini's_theorem
Japanese mathematician (1949–2020)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Yasumasa_Kanada
German mathematician (1690–1764)
solutions to Goldbach. Also, in 1729 Goldbach closely approximated the Basel problem, which prompted Euler's interest and concurring breakthrough solution
Christian_Goldbach
Index of lists with the same name
functions Apéry's constant - the solution to ζ(3) Artin conjecture Basel problem boils down to ζ(2) Birch and Swinnerton-Dyer conjecture Riemann hypothesis
List_of_zeta_functions
Mathematical power series of arctangent
Madhava–Gregory–Leibniz series: Benko, David; Molokach, John (2013). "The Basel Problem as a Rearrangement of Series". College Mathematics Journal. 44 (3):
Arctangent_series
Cartwright, but that she had not traced its origin. It still remains on the 4th problem sheet today for the Analysis IA course at Cambridge University. Consider
Proof_that_pi_is_irrational
Function that quantifies how near a number is to being rational
π". In Chudnovsky, David V.; Chudnovsky, Gregory V. (eds.). The Riemann Problem, Complete Integrability and Arithmetic Applications. Lecture Notes in Mathematics
Irrationality_measure
(2)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}} (the Basel problem) ζ ( 4 ) = ∑ k = 1 ∞ 1 k 4 = π 4 90 {\displaystyle \zeta (4)=\sum _{k=1}^{\infty
List_of_mathematical_series
Japanese mathematician and cartographer
(發微算法演段諺解) OCLC 22056085721 Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in shinto shrines Soroban, a Japanese
Takebe_Kenkō
Banking regulation framework
Basel II is a framework on banking regulations published by the Basel Committee on Banking Supervision in June 2004 to determine the minimum capital that
Basel_II
1970 book by Petr Beckmann
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
A_History_of_Pi
Varying methods used to calculate pi
value of Pi". Rather, the bill dealt with a purported solution to the problem of geometrically "squaring the circle". The bill was nearly passed by the
Approximations_of_pi
Mathematical criterion about whether a series converges
diverges. The case of p = 2 , k = 1 {\displaystyle p=2,k=1} is the Basel problem and the series converges to π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}}
Convergence_tests
1897 event in Basel, Switzerland
congress of the Zionist Organization (ZO) held in the Stadtcasino Basel in the city of Basel on August 29–31, 1897. Two hundred and eight delegates from 17
First_Zionist_Congress
English mathematician (1686–1751)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
John_Machin
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
ISBN 978-0-201-89683-1. Ed Sandifer, How Euler Did It — Estimating the Basel problem Archived 2005-05-13 at the Wayback Machine (2003) Paule, Peter; Schneider
Harmonic_number
the primes diverges Banach fixed-point theorem Banach–Tarski paradox Basel problem Bolzano–Weierstrass theorem Brouwer fixed-point theorem Buckingham π
List_of_mathematical_proofs
theory) Barban–Davenport–Halberstam theorem (analytic number theory) Basel problem (mathematical analysis) Beatty's theorem (Diophantine approximation)
List_of_theorems
Japanese mathematician (c. 1642–1708)
in Japan. He successfully applied it to problems suggested by his contemporaries. Before him, these problems were solved using arithmetical methods. In
Seki_Takakazu
Sum of the inverses of the positive cubes
of computers and to algorithmic improvements. Riemann zeta function Basel problem — ζ(2) Catalan's constant List of sums of reciprocals Wedeniwski (2001)
Apéry's_constant
Proofs of Fermat's theorem on sums of two squares Riemann zeta function Basel problem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture
List_of_number_theory_topics
Theorem in transcendental number theory
function j was conjectured by Daniel Bertrand in 1997, and remains an open problem. Writing q = e2πiτ for the square of the nome and j(τ) = J(q), the conjecture
Lindemann–Weierstrass_theorem
Mathematical series with a finite sum
The reciprocals of square numbers produce a convergent series (the Basel problem): 1 1 + 1 4 + 1 9 + 1 16 + 1 25 + 1 36 + ⋯ = π 2 6 . {\displaystyle
Convergent_series
Chinese mathematician-astronomer (429–500)
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Zu_Chongzhi
Texts belonging to the Śrauta ritual
similar interest and approach to doubling and other geometric transformation problems. Abraham Seidenberg, followed by Bartel Leendert van der Waerden, sees
Shulba_Sutras
Mathematical problem
DocCourse). Basel: Birkhäuser. pp. 1–86. ISBN 978-3-7643-8961-1. Zbl 1221.20045. Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the
Zero-sum_problem
Swiss mathematician (1667–1748)
born in Basel, the son of Nicolaus Bernoulli, an apothecary, and his wife, Margaretha Schonauer, and began studying medicine at University of Basel. His
Johann_Bernoulli
Mathematician
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
John_Wrench
Mnemonics of pi's digits
who could estimate its value? For me, your problem had equal advantages. Long ago, mysterious, a problem blocked All the admirable process, the great
Piphilology
Zoo in Basel, Switzerland
Zoo Basel is a zoological garden in the city of Basel, Switzerland. Its official name is Zoologischer Garten Basel — or in English: Basel Zoological Garden
Zoo_Basel
} Therefore, 22/7 > π. The evaluation of this integral was the first problem in the 1968 Putnam Competition. That the integral is positive follows from
Proof_that_22/7_exceeds_π
Italian mathematician (1626–1686)
Society. Mengoli died in Bologna in 1685. Mengoli first posed the famous Basel problem in 1650, solved in 1735 by Leonhard Euler. In 1650, he also proved that
Pietro_Mengoli
3rd century calculation of π by Liu Hui
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Liu_Hui's_π_algorithm
Russo–Vallois integral Stratonovich integral Skorokhod integral Miscellaneous Basel problem Euler–Maclaurin formula Gabriel's horn Integration Bee Proof that 22/7
Common integrals in quantum field theory
Common_integrals_in_quantum_field_theory
1735—Leonhard Euler solves the Basel problem, relating an infinite series to π. 1736—Leonhard Euler solves the problem of the Seven Bridges of Königsberg
Timeline_of_mathematics
A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Basel problem Six nines in π Other topics related to π Tau v t e
Madhava's_correction_term
Misunderstanding in Japanese education
until you are satisfied,' but if teachers are like this, it must be a problem before educational reform." Other issues with treating pi as approximately
Pi_is_3
Fastest curve descent without friction
Springer Basel Aktiengesellschaft. pp. 117–118. ISBN 978-3-0348-5068-1. Babb, Jeff; Currie, James (July 2008), "The Brachistochrone Problem: Mathematics
Brachistochrone_curve
Ratio of the perimeter of Bernoulli's lemniscate to its diameter
{3}}}}{\dfrac {\varpi }{12^{1/8}\pi }}.} In a spirit similar to that of the Basel problem, ∑ z ∈ Z [ i ] ∖ { 0 } 1 z 4 = G 4 ( i ) = ϖ 4 15 {\displaystyle \sum
Lemniscate_constant
Group of Vedic Sanskrit texts
square produces an area double the size of the original square. Another problem tackled by Baudhāyana is that of finding a circle whose area is the same
Baudhayana_sutras
subsequence with respect to the sup norm. Bargmann Bargmann transform Basel The Basel problem says ∑ 1 ∞ 1 n 2 = π 2 6 . {\displaystyle \sum _{1}^{\infty }{\frac
Glossary of real and complex analysis
Glossary_of_real_and_complex_analysis
Leonhard Euler solves the Basel problem, first posed by Pietro Mengoli in 1644, and the Seven Bridges of Königsberg problem. May 22 – George Hadley publishes
1735_in_science
One over a whole number
{1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}.} The Basel problem concerns the sum of the square unit fractions: 1 + 1 4 + 1 9 + 1 16
Unit_fraction
Theorem in number theory
_{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}} (see the Basel problem), the above constant log 5/3 = 0.51082... can be improved to log π2/6
Divergence of the sum of the reciprocals of the primes
Divergence_of_the_sum_of_the_reciprocals_of_the_primes
quasilinear parabolic problems) and went to Besançon with a post-doc fellowship. He then spent two years as an assistant in Basel, where he received his
Joachim_Escher
Criterion for the convergence of a series
series (1 + 1 + 1 + 1 + ⋯) diverges, the second (the one central to the Basel problem) converges absolutely and the third (the alternating harmonic series)
Ratio_test
Type of figurate number constructed by combining heptagons
x={\frac {5n^{2}-3n}{2}}} for its unique positive root n. "Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers" (PDF). Archived from the original
Heptagonal_number
first-order ordinary differential equations, 1735 - Leonhard Euler solves the Basel problem, relating an infinite series to π, 1736 - Newton's Method of Fluxions
Timeline of calculus and mathematical analysis
Timeline_of_calculus_and_mathematical_analysis
several significant events in mathematics, technology, and medicine. The Basel problem is posed by Pietro Mengoli, and will puzzle mathematicians until solved
1644_in_science
Construction in functional analysis, useful to solve differential equations
}|f(x)|^{2}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}.} Known as the Basel problem, this series converges to π 2 6 {\textstyle {\frac {\pi ^{2}}{6}}}
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Persian astronomer and mathematician (c. 1380–1429)
described him as "a remarkable scientist" who "could solve the most difficult problems". Al-Kashi produced a Zij entitled the Khaqani Zij, which was based on
Jamshid_al-Kashi
Calendar year
Leonhard Euler solves the Basel problem, first posed by Pietro Mengoli in 1644, and the Seven Bridges of Königsberg problem. The King's Highway (Charleston
1735
Maclaurin series of sine, cosine and arctangent
Madhava–Gregory–Leibniz series: Benko, David; Molokach, John (2013). "The Basel Problem as a Rearrangement of Series". College Mathematics Journal. 44 (3):
Madhava_series
Swiss mathematician (1655–1705)
generational suffix I, as in Jacob I Bernoulli. Jacob Bernoulli was born in Basel in the Swiss Confederation; the son and grandson of Protestant spice merchants
Jacob_Bernoulli
travel, tourism, insurance
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